I also purchased a desoldering pump. Figured it's something I would eventually need anyway.
As a consequence, I was now able to successfully desolder both ROM chips from the cartridge.
I was also able to dump them using my Arduino program. There was a slight problem in doing so: The legs of the ROM chips are short, and if the ROM chip is inserted into a standard issue breadboard, the chip just does not make proper contact with the springs and it just pops out. So during the dumping I had to press hard on the chip with both thumbs to ensure it reads all bits properly. Even then, I had to dump it several times, because I saw missing bits / discrepancies between successive dumps. Finally, I added in the ROM dumper some code that reads the ROM several times and chooses the result that has most bits on (and indicates if there were discrepancies). Coincidentally, I finally got them both without discrepancies.

So now I have two 65536 byte files containing the respective ROM data from both chips.
([EDIT: The PRG ROM is actually 128 kilobytes, so 128+64 it should be. I underdumped. I figured this out and perfectly fixed it later. This is an older post.])

Now, I need to figure out what kind of mapper this board has. I mean, I can see it has two 74LS161As (4-bit binary counter) and one 74LS153 (dual 4-input multiplexer) (see photos in the first post), but that does not tell me much. I created a dummy test ROM by just putting a 16-byte header, the 65536-byte PRG ROM and the 65536-byte CHR ROM in a sequence and running through 256 iNES mapper numbers and seeing how they work in FCEU. From those tests (collectively), I was able to get a properly functioning main menu (with links leading nowhere), a properly working Urban Champion game, a properly working Legend of Kage with broken graphics, and a semi-properly working Galaxian with utterly broken graphics. Work in progress...

Taking the ROM page that has the main menu program only, and tracing it using my own emulator, I found out which memory writes it does when a game is chosen.
At reset: Byte $00 is written to $FF00, and $00 is written to $FF00 (again).
Choose game 0: Byte $00 is written to $FFA5, $00 to $FF00 and $E2 to $FF03.
Choose game 1: Byte $00 is written to $FFC8, $00 to $FF00 and $E2 to $FF03.
Choose game 2: Byte $00 is written to $FF10, $00 to $FF00 and $E2 to $FF03.
Choose game 3: Byte $00 is written to $FFED, $00 to $FF00 and $E2 to $FF03.
Choose game 4: Byte $00 is written to $FF03, $00 to $FF00 and $E2 to $FF03.
From this, it looks like it uses the address, rather than the data, to choose a function. (And indeed, the data lines from the ROM chips are directly wired into the cartridge's data lines without intervention if I read right.) However, what those functions are is so far a bit unclear. It looks like the low 4 bits choose VROM page and the next 4 might bits choose a ROM page, or vice versa, but this is mere guessing.
Also, what kind of mapping functions does it provide for the games themselves? I think at least Goonies ought to use more than one VROM page.
EDIT: I traced (some of) the pinouts on the cartridge to help figuring out what the mapper does.
Code:
ROM CHIP 1:
A15 1 IC1.PIN12 28 +5V
2 PRG A12 A14 27 IC1.PIN14
3 PRG A7 & IC1.PIN3 26 PRG A13
4 PRG A6 & IC1.PIN4 25 PRG A8
5 PRG A5 & IC1.PIN5 24 PRG A9
6 PRG A4 & IC1.PIN6 23 PRG A11
7 PRG A3 & IC2.PIN3 /OE 22 IC1.PIN13
8 PRG A2 & IC2.PIN4 21 PRG A10
9 PRG A1 & IC2.PIN5 /CE 20 IC1.PIN2
10 PRG A0 & IC2.PIN6 19 PRG D7
11 PRG D0 18 PRG D6
12 PRG D1 17 PRG D5
13 PRG D2 16 PRG D4
14 GND 15 PRG D3
Other:
Cart's PRG A14 = IC3.PIN3,4
where IC1, IC2 are GD74LS161A
IC3 is GD74LS153
And IC1.PIN2 -> IC2.PIN2
IC2.PIN10 -> IC3.PIN11
IC3.PIN13 -> IC3.PIN15
Plus many lines that go under the ICs and are thereby efficiently obscured.
ROM CHIP 2:
A15 1 IC2.PIN14 28 +5V
2 CHR A12 A14 27 IC2.PIN13
3 CHR A7 A13 26 IC2.PIN12
4 CHR A6 25 CHR A8
5 CHR A5 24 CHR A9
6 CHR A4 23 CHR A11 & IC3.PIN15
7 CHR A3 /OE 22 CHR /RD
8 CHR A2 21 CHR A10 & IC3.PIN12,10
9 CHR A1 /CE 20 CHR /A13
10 CHR A0 19 CHR D7
11 CHR D0 18 CHR D6
12 CHR D1 17 CHR D5
13 CHR D2 16 CHR D4
14 GND 15 CHR D3